Slice the white bread or Ciabatta in cubes. In a small saucepan on the stove top, heat oil on medium-high heat.
Add bread cubes and fry until golden brown (it only takes about 30 seconds, so watch them carefully). Drain on kitchen paper and set aside to cool.
Combine mayonnaise, vinegar, garlic, mustard and black pepper in a bowl. Whisk until well blended. Taste and add more mayonnaise, red wine vinegar and/or black pepper as desired.
Drizzle in the oil and add 1/2 cup of the grated or shaved Parmesan cheese.
Pour the dressing over the lettuce and toss gently with the lemon juice if using. Scatter the remaining Parmesan cheese, add the Croutons (white bread or Ciabatta cubes) and crack more black pepper over top, and serve.